reserve i,j for Nat;
reserve A,B for Ring;
reserve K, L for Field;

theorem Th83:
  for x,a be Element of F_Complex st  a <> 0.F_Complex &
  a in the carrier of FQ_Ring(x)
  ex g be Element of Polynom-Ring F_Rat
  st not g in Ann_Poly(x,F_Rat) & a = hom_Ext_eval(x,F_Rat).g
  proof
    let x,a be Element of F_Complex;
    set M = {p where p is Polynomial of F_Rat:Ext_eval(p,x)=0.F_Complex};
    assume that
A1:  a <> 0.F_Complex and
A2:  a in the carrier of FQ_Ring(x);
     consider g be Element of Polynom-Ring F_Rat such that
A3:  a = hom_Ext_eval(x,F_Rat).g by A2,Lm62;
     take g;
     thus not g in Ann_Poly(x,F_Rat)
     proof
       assume g in Ann_Poly(x,F_Rat);
       then consider g1 be Polynomial of F_Rat such that
A5:    g1 = g and
A6:    Ext_eval(g1,x)=0.F_Complex;
       thus contradiction by A1,A6,A3,A5,Def9;
     end;
     thus thesis by A3;
  end;
